The BiConjugate Gradient method often displays rather irregular
convergence behavior. Moreover, the implicit
decomposition of the
reduced tridiagonal system may not exist, resulting in breakdown of the algorithm. A related algorithm, the
Quasi-Minimal Residual method of Freund and
Nachtigal [102], [103]
attempts to overcome
these problems. The main idea behind this algorithm is to solve the
reduced tridiagonal system in a least squares sense, similar to the
approach followed in GMRES.
Since the constructed basis for the
Krylov subspace
is bi-orthogonal , rather than orthogonal as in GMRES,
the obtained solution is viewed as a quasi-minimal residual solution,
which explains the name. Additionally, QMR uses look-ahead techniques
to avoid breakdowns in the underlying Lanczos process, which makes it
more robust than BiCG.
Figure: The Preconditioned Quasi Minimal Residual Method
without Look-ahead